Some similarities classes related to O(2,1) in split quaternions

Abstract

In this paper, we show that PO(2, 1) can be represented by invertible split quaternions. We introduce the concepts of t-similarity and t-pseudosimilarity of split quaternions. By such two similarities, we obtain several equivalence relations in split quaternions. We also show that a and b are semisimilar if and only if a² and b² are similar.

Keywords:

• Split quaternion
• t-similar
• t-pseudosimilar
• semisimilar
• O(2, 1)

2010 Mathematics Subject Classifications:

• Primary 15A24
• Secondary 15A33

Acknowledgments

The authors would like to express their deep gratitude to the referee and the handling editor for carefully reading this paper and some useful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by Natural Science Foundation of China [grant number 11871379], the Innovation Project of Department of Education of Guangdong Province [grant number 2018KTSCX231] and Key project of National Natural Science Foundation of Guangdong Province Universities [grant number 2019KZDXM025].